Tautology, Valid, Contingent, Unsatisfiable, Contradiction: relationship?

Question

I am trying to clear my doubts about various terms: tautology, contradiction, contingent, satisifiable, unsatisfiable, valid and invalid. I have read on them from various sources and was thinking about them. I am putting all my understanding below in points. Some points are definitions straight up from other sources, so they must be correct. Some points I have prepared from understanding I got after reading and thinking. I need confirmation whether they (below text in quotes / highlights) are correct or not.

These are definitions:

1. Tautology: a formula or assertion that is true in every possible interpretation (that is, for all assignment of values to its variables). Ref
2. Contradiction: a formula or assertion that is false in every possible interpretation.
3. A formula that is neither a tautology nor a contradiction is said to be logically contingent.
   Such a formula can be made either true or false based on the values assigned to its propositional variables.
4. A formula is satisfiable if it is true under at least one interpretation.
   So, its either contingent or tautology. Ref
5. If a logic is a contradiction then it is said to be unsatisfiable.
6. A formula is valid only if it is tautology. Ref
7. A formula is invalid if it is contradiction or contingent.

Based on these definitions, I tried preparing diagram highlighting how these concepts overlap:

Based on this diagram, I tried to give answers to some problems.

For example, what is negation of tautology? From above diagram, I felt that it could be either contradiction or contingent. But seems that I was wrong. Above diagram means:

"Given an assertion, if it is not tautology, it can be either contradiction or contingent."

But it does not means:

"Negation of assertion which is tautology can be contingent or contradiction"

I asked this doubt earlier. As pointed out by J.G. in his comment, that I was simply negating definitions above, where I should have actually tried investigating how model (set of values assigned to variables of formula) satisfying given formula behaves for negation of that formula. It took a while for me to grasp that. However I feel I understand it now. I have come up with following relations of nature of any given assertion and nature of its negation:

Purpose of above table: Given any assertion, above table gives nature of its negation.
Purpose of below table: Given fact that certain assertion is not of certain nature, below table tells what could be possible nature of that assertion.
(I have given examples in brackets to support the facts.)

Can someone please confirm that my understanding which I put above in various points in yellow highlights / quotes is indeed correct. I dont know if I am overthinking. But I just want exhaustive understanding and know all possibilities, especially because in exam, they ask truthfulness of subtle fact and check out understanding. So trying to know and think exhaustively beforehand.

Answer 1

The notions and their negations will get clearer once you realize the meta-logical quantifications they involve, and how these quantifications systematically behave under negation.

All the definitions you listed quantify over interpretations: A statement is valid iff it is true under all interpretations, satisfiable iff it is true under some interpretation, contradictory iff it is valid under no interpretation, and so on.

In general, we have

• "not all interpretations" = "there exists some interpretation such that not";
• "not some interpretation" = "for all interpretations not";
• "not no interpretation" = "there exists some interpretation such that".

So let's apply these equivalences to each of the definitions*:

-------------------------------------------------------------------------------------------------------------------------    
Notion          Definition                           Negation of definition                  Negation of notion    
-------------------------------------------------------------------------------------------------------------------------
tautological    all i. true (= no i. false)          not all i. true (= some i. false)       contradcictory or contingent
contradictory   all i. false (= no i. true)          not all i. false (= some i. true)       satisfiable
(= unsatisfiable)
contingent      some i. true and some i. false       not (some i. true and some i. false)    contradictory or tautological
                                                     = (not some i. true) or (not some i. false)
                                                     = no i. true or no i. false
                                                     = all i. false or all i. true
satisfiable     some i. true (= not all i. false)    no i. true (= all i. false)             contradictory
unsatsifiable   no i. true (= all i. false)          some i. true (= not all i. false)       satisfiable
(= contradictory)
valid           (see tautological)
(= tautological)
invalid         not valid = not all i. true          all i. true                             tautological

So we have

---------------------------------------------------------------------
Notion              Negation               can but doesn't have to be
---------------------------------------------------------------------
not tautological    = not valid         
                    = invalid
                    = contradictory        unsatisfiable (if contradictory),
                      or contingent        satisfiable (if contingent)
not contradictory   = not unsatisfiable    tautological,
                    = satisfiable          contingent,
                                           invalid
not contingent      = contradictory        unsatisfiable, invalid (if contradictory),
                      or tautological      satisfiable (if tautological)
not satisfiable     = unsatisfiable        invalid (must be)
                    = contradictory
not valid           = invalid

                    = not tautological
                    = contradictory        unsatisfiable (if contradictory),
                      or contingent        satisfiable (if contingent)
not invalid         = valid                satisfiable (must be)
                    = tautological

The problem with the negations your first table is that your negation is to strong: The negation of "all interpretations" is just "not all interpretations", i.e. "there are some interpretations such that not", and not (as you did) "no interpretation". So the negation of "valid" is just "not true under all interpretations", which can be contradictory or contingent, and not "true under no interpretation", which would be contradictory. Likewise, the negation of contradictory (= false under all interpretations) is just "not false under all interpretations", i.e. "true under some interpretations", which satisfiable, and not the stronger statement "true under all interpretations", which would be tautological.

The diagram you made is correct, and explicates the misunderstanding as follows: Negation does not mean opposite. Negating a notion does not get you to the other extreme of the diagram, only to the complementary half, i.e. the entire part not covered by that notion: "not contradictory" gives you everything in the range of "satisfiable", not just the extreme "tautology". "not tautological" just gives you "invalid", not the opposite "contradictory". "Not satisfiable" is "contradiction", not "invalid", "not invalid" is "tautology", not "satisfiable", and lastly, if something is "not contingent" it must be in either "contradiction" or "tautology". That covers all the possible cases.

* Note that in the case of first-order logic, sometimes a distinction is made between a formula that is valid (true in all first-order interpretations) and one that is tautological (a first-order instance of a propositional tautology, i.e. one that has the form of a tautological propositional formula but with predicate logical formulas in the place of propositional variables). All tautological formulas are also valid, e.g. $\forall x P(x) \lor \neg \forall x P(x)$, but not all first-order validities are merely instances of propositional tautologies, e.g. $\forall x (x = x)$.

Answer 2

(Please refer to point #4 below for a refinement of the above diagram.)

1. • Every formula that's not valid is invalid.
   • Every formula that's not satisfiable is unsatisfiable.

   On the other hand,

   • The negation of every valid formula is unsatisfiable (not merely invalid).
   • The negation of every satisfiable formula is invalid (not necessarily unsatisfiable).

2. Equivalently:

   • Valid and Invalid are complementary sets.
   • Satisfiable and Unsatisfiable are complementary sets.
   • The negation of every unsatisfiable formula is valid (not merely satisfiable).
   • The negation of every invalid formula is satisfiable (not necessarily valid).

3. To wit:

   • $\big(\forall x\,P(x)\big)$ and its negation $\big(\exists x\,¬P(x)\big)$ are both satisfiable.
   • $\big(\forall x\,P(x)\big)$ and its negation $\big(\exists x\,¬P(x)\big)$ are both invalid.
   • $(\forall x\:x=x)$ is valid, and its negation $(\exists x\,x\ne x)$ is unsatisfiable, not merely invalid.
   • $(\exists x\,x\ne x)$ is unsatisfiable formula, and its negation $(\forall x\:x=x)$ is valid, not merely satisfiable.

4. 1. Tautology: a formula or assertion that is true for all assignment of values to its variables
   2. Contradiction: a formula or assertion that is false in every possible interpretation.
   3. A logically contingent formula can be made either true or false based on the values assigned to its propositional variables.

   Notice that collectively, these three terms are being defined in terms of a formula's truth-functional form; as such, in first-order logic, the contingent formulae actually form a proper superset of the satisfiable but invalid formulae; in this case:

   • $(\forall x\:x=x)$ is contingent and (first-order) valid.

   • $(\exists x\,x\ne x)$ is contingent and (first-order) unsatisfiable.

   • These corollaries are false:

     6. A formula is (first-order) valid only if it is tautology.
     7. A formula is (first-order) invalid if it is contradiction or contingent.

Related: Complement versus Negation